Phase Modulation

The phase modulation is similar to the frequency modulation and is an important technique in digital communication systems.

We've all heard the radio AM and FM radio. But the phase modulation appears to be in a different category: “Radio PM ” It is not a common term. It appears that the phase modulation is more relevant in the context of digital RF. In a sense, But, we can say that AM radio is as common as the FM radio simply because there is little difference between the phase modulation and frequency modulation. FM and PM are considered as two closely related variants angle modulation , dove “angle” It refers to the modification of the amount passed to a sine or cosine function.

In mathematics

We have seen in the preceding article that the frequency modulation is obtained by adding the integral of the in-band based on the argument of a sine or cosine function signal (where the sine or cosine function represents the carrier):

recall, however, we introduced frequency modulation first discussing the phase modulation: adding the baseband signal itself, instead of the integral of the baseband signal, the phase varies according to the value of the base band. Therefore, the phase modulation is actually a bit 'easier than the frequency modulation

As with frequency modulation, we can use the modulation index for making changes in the most sensitive phase to changes in the value of the baseband:

The similarity between phase modulation and frequency modulation becomes clear if we consider a baseband signal at a single frequency. Let's say xbb(t) = no (ωBBt). The integral of the breast is negative cosine (plus a constant, that we can ignore here), in other words, the integral is simply a shifted version of the original signal in time. Therefore, if we execute the phase modulation and frequency modulation with this baseband signal, the only difference in the forms of modulated wave will be the alignment between the value of the base band and the variations in the carrier; the same variations are the same. This will be clearer in the next section, where we will see some graphs in the time domain.

It is important to note that it is instantaneous phase, just like the frequency modulation is based on the concept of instantaneous frequency. The term “phase” It is rather vague. A familiar meaning refers to the initial state of a sine wave; eg, sine wave “normal” starts with a value equal to zero and then increases towards its maximum value. A sine wave that starts at a different point in its cycle has a phase offset. We can also think of the stage as a specific part of a complete cycle of waveforms; for example, a phase π / 2, a sine wave has completed a quarter of its cycle.

These interpretations of “phase” We do not help us a lot when we're dealing with a phase which varies continuously in response to a waveform baseband. Rather, we use the concept phase snapshot, that is the phase at a given moment, which corresponds to the value passed (at any given time) to a trigonometric function. We can think of these continuous variations in the instantaneous phase as “pushing” the carrier value farther or closer to the previous state of the waveform.

Another thing to keep in mind: trigonometric functions, including sine and cosine, They operate on the corners. Change the subject of a trigonometric function is equivalent to change the angle, and this explains why both the FM that the PM are described as the angle modulation.

The time domain

We will use the same waveforms that we used for the discussion FM, ie a carrier 10 MHz and a sinusoidal signal in base band by 1 MHz:

Here the shape FM band (con m = 4) we saw in the previous article:

We can calculate the waveform of PM using the following equation, in which the carrier wave signal added to the subject uses the positive breast (that is, the original signal) instead of the negative cosine (ie the integral of the original signal).

Here is the graph of PM:

Before discussing, we also observe a graph that shows the shape of FM wave and the waveform of PM:

The first thing that comes to mind is that, from a visual point of view, FM is more intuitive PM: there is a clear visual connection between the sections with the highest and lowest of the modulated waveform frequency and the highest and lowest values ​​baseband. With PM, the relationship between the waveform of the base band and the behavior of the carrier is perhaps not immediately apparent. However, after a while’ inspection we can see that the carrier frequency corresponds to PM slope the waveform of the base band; the highest occurring during the steepest positive slope of sections of xbb frequency and the lower frequency sections occur during the steepest negative slope.

Does this make sense: Remember that the frequency (in function of time) It is the derivative of phase (in function of time). With the phase modulation, the slope of the baseband signal determines the speed at which the phase changes and the speed with which the phase change is equivalent to the frequency. Therefore, a waveform PM, the high slope of the base band corresponds to the high frequency and the low slope of the base band corresponds to the low frequency. With frequency modulation, we use the integral of xbb, which has the effect of moving the supporting sections of high-frequency (or low) on the baseband values following the slope portions high (or low) the waveform of the base band.

The frequency domain

The previous diagrams in the time domain demonstrate what has been said previously: frequency modulation and phase modulation are quite similar. not surprisingly, then, that the PM effect in the frequency domain is similar to that of FM. Here are spectra for the phase modulation with the carrier signals and base band used above:

With this third article I finished the description of analog modulation, it remains to describe the techniques used for modulation with digital signals

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